In the form of a bound on the determinant of a symmetric positive definite matrix, Hadamard's result gained
importance due to connections to Fredholm's theory of differential equations, with a new proof by Wirtinger
in 1907 [1]} and a generalisation by Fischer in 1908 [2]}. In fact, we shall have use for Fischer's
inequality in later sections of this paper, and so provide a proof modeled closely on the original. Both results in this section are easily established using techniques of positive definite matrices. Bechenbach and Bellman claim that there are perhaps a hundred proofs of the Hadamard inequality [3]}; a one-line proof is given on page 505 of Horn and Johnson [4]}.
